Trapdoor Functions from the Computational Diffie-Hellman Assumption

Trapdoor functions (TDFs) are a fundamental primitive in cryptography. Yet, the current set of assumptions known to imply TDFs is surprisingly limited, when compared to public-key encryption. We present a new general approach for constructing TDFs. Specifically, we give a generic construction of TDFs from any Chameleon Encryption (Dottling and Garg [CRYPTO'17]) satisfying a novel property which we call recyclability. By showing how to adapt current Computational Diffie-Hellman (CDH) based constructions of chameleon encryption to yield recyclability, we obtain the first construction of TDFs with security proved under the CDH assumption. While TDFs from the Decisional Diffie-Hellman (DDH) assumption were previously known, the possibility of basing them on CDH had remained open for more than 30 years.